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A manufacturer estimates that its product can be produced at a total cost of C(x) = 45,000 + 100x + x3 dollars. If the manufacturer's total revenue from the sale of x units is R(x) = 4000x dollars, determine the level of production x that will maximize the profit. (Round your answer to the nearest whole number.)

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TaeKwonDoIsDead

Answer:

x = 1950 units

Step-by-step explanation:

We need to maximize Profit (P(x)). We know:

Profit = Revenue -  Cost

So,

[tex]P(x)=R(x)-C(x)\\P(x)=4000x-[45000+100x+x^3]\\P(x)=4000x-45000-100x-x^3\\P(x)=-x^3+3900x-45000[/tex]

This follows quadratic equation of the form  [tex]ax^2+bx+c[/tex]

So, matching, we have:

a = -1

b = 3900

c = -45,000

The max occurs at the value  [tex]x=-\frac{b}{2a}[/tex]

So, the level of production, x , that will maximize profit is:

[tex]x=-\frac{b}{2a}\\x=-\frac{3900}{2(-1)}\\x=1950[/tex]

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